Lego Cable Stayed Bridge
A cable stayed bridge is one of the more elegant bridge designs. Here's some information on cable stayed bridges from Wikipedia; since it's online, it must be true. My own inspiration for this project is the Leonard P. Zakim Bunker Hill Memorial Bridge (on right, picture also from Wikipedia), which was constructed just around the time we were leaving Boston. Wouldn't it be neat to construct a Lego model of the bridge? Of course, many people have already built cable-stayed bridges. Most of them require more Legos than we have (even with 90+ gallons). Besides, I'm not really a model-builder: I don't have the patience to recreate something exactly. What I'm interested in is the technical challenge of reproducing the components; others can put them together. In particular, the cables intrigued me: How could you do Lego cables that hang obliquely? The easy solution is to use string. But that feels like cheating. An elegant solution would use Lego bricks for all components. The MathematicsOne solution (as anyone who's ever played with Legos will know) is this: Take two 1 by n flats or bricks, and join them by overlapping the last peg. The two pieces can then swivel about their ends. You can attach their ends to a third piece (again by overlapping the last peg) and form a triangle. The challenge is to find three pieces that fit together to form a right triangle. Since the Lego pegs are the only valid attachment points, and they are at fixed distances, the mathematical question can be phrased as follows:
Find three whole numbers A, B, and C that can form the sides and hypotenuse of a right triangle.
Such a set is known as a Pythagorean triplet. For example, (3, 4, 5) is the simplest and best-known Pythagorean triplet. By using the Lego pegs as distance markers, it's clear that you can construct slants of arbitrary size. The picture shows a 3-4-5 right triangle constructed using 1 by n flats. There is a mechanical limit to the allowable angle between the two flats (essentially when the top flat overlaps the remaining pegs on the bottom flat); to avoid this, you can put a 1 by 1 peg in between the two flats that form the sides of the triangle.
For this project, I used the following primitive Pythagorean triplets: 3-4-5, 8-15-17, and 5-12-13. Since these by themselves would overlap, I scaled them (and flipped them), and ultimately used the following, where the first number indicates the vertical height and the second number the horizontal distance:
4-3-5 for the lowest.
6-8-10 for the next (the 3-4-5, doubled)
8-15-17 for the third,
10-24-26 for the last (the 5-12-13, doubled)
The reason for this is that the four "cables" would then have four different slopes. The one closest to the base would be the steepest, with gradually decreasing slopes as you reached the top. (Lego purists take note: the collection included some Megabloks, in part because for a while, you couldn't find brick collections with a decent number of ordinary bricks. If the piece was the right color and shape, I used it) The picture shows the set-up, from bottom to top; you should be able to figure out how the central tower was constructed, with attachment points at heights 4, 6, 8, 10, and 12.
The support beam for the road deck is constructed in a similar manner. In this case, attachment points were placed at positions 3, 8, 15, and 24. The picture shows the original placement; when I went to place the road deck, I decided that the attachment points needed to extend a little further from the beam.
Here are a few pictures of the final bridge. One neat feature is that this bridge actually works! What I mean is this: the bridge deck is actually supported by the "cables", so if you drove an actual vehicle over the bridge, it would shatter into pieces of broken plastic. But if the plastic was actually strong enough to allow the cables to support the tension and the central pylon to support the compression, the bridge would work as it's supposed to.
Extending the Bridge
This only gives four cables. The classic method of generating Pythagorean triplets is the following: Let A be an odd number. Square it, subtract one, and halve it; this gives you B. Add one, which will give you C. For example, take 7. The square of 7 is 49; subtract 1 to get 48, then take half of 48 to get 24, which is B; adding 1 to 24 gets 25, which is C. This gives you the triplet (7, 24, 25).
A second method starts with an even number. Let A be an even number. Take half of A and square it; then subtract 1 to get B and add 1 to get C. For example, if A = 12, half of 12 is 6; squared is 36, and subtracting 1 gives you 35. Adding 1 (to 36) gives you 37, for the triplet (12, 35, 37). Of course, the problem is that this places the endpoint of the fifth cable all the way out at peg 35 (leaving the space between peg 24 and 35 free of supports).
As an exercise (I am a math teacher, after all): Figure out how to use the Pythagorean triplets to give you a gradually decreasing slope. Lego constraints require that the distances differ by at least 2 units (so the attachment points on our central tower occurred at 4, 6, 8, and 10).
Another Method
The pictures above show the "fan" design for the cable stayed bridge. Another design is the "harp." Mathematically, this is a lot easier, since all you have to do is scale a triangle. For example, if you use the 3-4-5 triangle, put the attachment points at 3, 6, 9, 12, etc. along the central tower, and at 4, 8, 12, 16, etc. along the road deck.
Back to Lego Mathematics.